The weekly, four-weekly and thirteen-weekly options were introduced in April 2010.
It's been years since I was involved with State Pensions but when I started, pensions were normally paid weekly by order book or four-weekly into an account.
Some folk chose to have it go in every thirteen weeks but those who were entitled only to an amount below a specified amount, had to be paid annually.
I'm not sure why you were given the option to be paid weekly as the default would normally have been a four-weekly payment.
In theory, the Secretary of State (in the form of the person processing the claim) can allocate any pay day and period but in practice that doesn't happen.
I assume there must be criteria for agreeing to pay weekly rather than four-weekly, based on cost implications but I don't know what they are.